ON v-MAROT MORI RINGS AND C-RINGS

Geroldinger, Alfred; Ramacher, Sebastian; Reinhart, Andreas

doi:10.4134/jkms.2015.52.1.001

Cited by 26 publications

(12 citation statements)

References 21 publications

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“…The property of being a Krull domain is a purely multiplicative one. Indeed, a domain R is a Krull domain if and only if its multiplicative monoid of nonzero elements is a Krull monoid (this characterization generalizes to rings with zero-divisors, see [45,Theorem 3.5]). If R is a Krull domain, then C(R [27]).…”

Section: Finitely Generated Monoidsmentioning

confidence: 99%

Factorization theory in commutative monoids

Geroldinger

Zhong

2020

Semigroup Forum

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This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

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Section: Finitely Generated Monoidsmentioning

confidence: 99%

Factorization theory in commutative monoids

Geroldinger

Zhong

2020

Semigroup Forum

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“…To give an example for a C-domain, let R be a Mori domain with nonzero conductor f = (R : R). If the class group C( R) and the factor ring R/f are both finite, then R is a C-domain by [19, Theorem 2.11.9] (for more on C-domains see [22,31]).…”

Section: Finitely Generated Monoids Of R-idealsmentioning

confidence: 99%

The monotone catenary degree of monoids of ideals

Geroldinger

Reinhart

2019

Int. J. Algebra Comput.

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Factoring ideals in integral domains is a central topic in multiplicative ideal theory. In the present paper we study monoids of ideals and consider factorizations of ideals into multiplicatively irreducible ideals. The focus is on the monoid of nonzero divisorial ideals and on the monoid of vinvertible divisorial ideals in weakly Krull Mori domains. Under suitable algebraic finiteness conditions we establish arithmetical finiteness results, in particular for the monotone catenary degree and for the structure of sets of lengths and of their unions.has factorizations with distance greater than M . However, at least those factorizations, which are powers of factorizations of a, can be concatenated, step by step, by factorizations whose distance is small and does not depend on M . This phenomenon is formalized by the catenary degree which is defined as follows. The catenary degree c(H) of H is the smallest N ∈ N 0 ∪ {∞} such that for each a ∈ H and each two factorizations z, z ′ of a there is a concatenating chain of factorizations z = z 0 , z 1 , . . . , z k+1 = z ′ of a such that the distance d(z i−1 , z i ) between two successive factorizations is bounded by N . It is well-known that the catenary degree is finite for Krull monoids with finite class group and for C-monoids (these include Mori domains R with nonzero conductor f = (R : R) for which the residue class ring R/f and the class group C( R) are finite).In order to study further structural properties of concatenating chains, Foroutan introduced the monotone catenary degree ([13]). The monotone catenary degree c mon (H) is the smallest N ∈ N 0 ∪ {∞} such that for each a ∈ H and each two factorizations z, z ′ of a there is a concatenating chain of factorizations z = z 0 , z 1 , . . . , z k+1 = z ′ of a such that the distance between two successive factorizations is bounded by N and in addition the sequence of lengths |z 0 |, . . . , |z k+1 | of the factorizations is monotone (thus either |z 0 | ≤ . . . ≤ |z k+1 | or |z 0 | ≥ . . . ≥ |z k+1 |). Therefore, by definition, we have c(H) ≤ c mon (H) and2010 Mathematics Subject Classification. 13A15, 13F15, 20M12, 20M13.

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“…For certain classes of weakly Krull C-monoids (including orders in number fields) the elasticity is even accepted whenever it is finite ([16, Theorem 4.4] but this is not true for C-monoids in general (see [12, page 226], [3], and the literature therein). We refer to [12,Chapter 2] for the basics on C-monoids and to [21,15,6] for recent progress and new classes of examples. Here we just mention the most significant example from ring theory.…”

Section: Preliminariesmentioning

confidence: 99%

On the Arithmetic of Mori Monoids and Domains

Zhong¹

2019

Glasgow Math. J.

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Let R be a Mori domain with complete integral closure R, nonzero conductor f = (R : R), and suppose that both v-class groups Cv(R) and Cv ( R) are finite. If R/f is finite, then the elasticity of R is either rational or infinite. If R/f is artinian, then unions of sets of lengths of R are almost arithmetical progressions with the same difference and global bound. We derive our results in the setting of v-noetherian monoids.

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ON v-MAROT MORI RINGS AND C-RINGS

Cited by 26 publications

References 21 publications

Factorization theory in commutative monoids

Factorization theory in commutative monoids

The monotone catenary degree of monoids of ideals

On the Arithmetic of Mori Monoids and Domains

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